Nuclearity and Trace in Monoidal Bicategories and Extended Conformal Field Theories

James Hefford

Nuclear and trace ideals provide a way of formalising, respectively, partial compact closure and partial traces in monoidal categories [1]. Hilbert spaces and bounded linear maps provide the canonical example of a category with a nuclear and trace ideal: the Hilbert-Schmidt maps and the trace class maps respectively.

In this talk, I will give a more modern formulation of nuclear and trace ideals in terms of profunctors and use this to categorify the notions to the setting of monoidal bicategories $\mathcal{B}$ making use of the theory of pseudocoends [4]. A trace 2-ideal is a monoidal sub-biprofunctor of the hom $T(-,-)\hookrightarrow\mathcal{B}(-,-):\mathcal{B}\relbar\joinrel\mapstochar% \joinrel\rightarrow\mathcal{B}$ together with a functor $\mathrm{tr}:\mathchoice{{\vbox{\hbox{$\textstyle\sim$}}\kern-7.3611pt}}{{\vbox% {\hbox{$\scriptstyle\sim$}}\kern-4.80557pt}}{{\vbox{\hbox{$\scriptscriptstyle% \sim$}}\kern-3.40279pt}}{{\vbox{\hbox{$\scriptscriptstyle\sim$}}\kern-2.98611% pt}}\!\int^{a}T(a,a)\rightarrow\mathcal{B}(i,i)$ , and a certain higher coherence cell ensuring the trace is monoidally well-behaved. Trace 2-ideals capture the part of a monoidal bicategory that permits a higher trace in the sense of the shadows of [5], and in the case that $T(-,-)=\mathcal{B}(-,-)$ , I will show that the coherences of the pseudocoend coincide with those of [5].

A nuclear 2-ideal for $\mathcal{B}$ consists of a monoidal sub-biprofunctor of the hom $N(-,-)\hookrightarrow\mathcal{B}(-,-)$ equipped with a monoidal pseudonatural isomorphism, $\upsilon:N(a^{\dagger},b)\cong\mathcal{B}(i,\overline{a}\otimes b)$ and invertible 2-cells witnessing the yanking equation, together with coherences including the swallowtail equation familiar from the theory of compact closed bicategories [2]. Indeed, when $N(-,-)=\mathcal{B}(-,-)$ then $\mathcal{B}$ is a compact closed bicategory.

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As the main example, I will show that the bicategory $\mathsf{2Hilb}$ of infinite dimensional 2-Hilbert spaces has a nuclear and a trace ideal categorifying the story for $\mathsf{Hilb}$ . Similarly, the bicategory $\mathsf{Bord}$ of conformal cobordisms permits a nuclear and a trace ideal and I will use this to give a formalisation of a once extended conformal field theory as a nuclear 2-functor, $Z:\mathsf{Bord}\rightarrow\mathsf{2Hilb}$ . This categorifies the result of [3] and formalises some results from [6].

[1] S. Abramsky, R. Blute, and P. Panangaden. Nuclear and trace ideals in tensored $*$ -categories. Journal of Pure and Applied Algebra, 143(1):3-47, 1999.
[2] J. Baez and L. Langford, Higher-dimensional algebra IV: 2-Tangles, Advances in Mathematics 180 (2003) 705-764.
[3] R. Blute, P. Panangaden, and D. Pronk. Conformal field theory as a nuclear functor. Electronic Notes in Theoretical Computer Science, 172:101-132, 2007.
[4] S. Bozapalides. Théorie formelle des bicatégories. PhD thesis, Université Paris 7, 1976.
[5] K. Ponto and M. Shulman. Shadows and traces in bicategories. Journal of Homotopy and Related Structures, 8:151-200, 2013.
[6] S. Stolz and P. Teichner. What is an elliptic object? In Topology, geometry and quantum field theory: proceedings of the 2002 Oxford symposium in honour of the 60th birthday of Graeme Segal, number 308, page 247. 2004.